Problem of the Month
(November 2013)

Suppose you have resistors of resistance 1 through n. By putting these resistors in series and parallel, how close you can come to creating something with a resistance of some famous mathematical constant like π, e, ϕ, or √2? (For those unfamiliar with physics, two resistors with resistances X and Y have total resistance X+Y if they are in series, and 1 / (1/X + 1/Y) if they are in parallel.) If you use any n integer resistors, how close can you get the total resistance to one of these constants?

ANSWERS

Dave Langers wrote a powerful javascript program to solve this problem, which he put here.

Here are the best known approximations using resistors 1-n:

π Using Resistors 1-n

R = 1
error = -2.14
R = 3
error = -.142
R = 11 / 4
error = -.392
R = 29 / 9
error = .0806
R = 119 / 38
error = -.0100
R = 22 / 7
error = .00126

R = 1043 / 332
error = -.0000264
R = 6723 / 2140
error = -.00000387
(George Sicherman)
R = 355 / 113
error = 2.67 × 10-7
(Jon Palin)

R = 110028 / 35023
error = 1.44 × 10-8
(George Sicherman)
R = 208341 / 66317
error = 1.22 × 10-10
(Dave Langers)
R = 10631071 / 3383975
error = 2.02 × 10-11
(Michael Moffitt)

e Using Resistors 1-n

R = 1
error = -1.72
R = 3
error = .282
R = 11 / 4
error = .0317
R = 50 / 19
error = -.0867
R = 19 / 7
error = -.00400
R = 106 / 39
error = -.000333

R = 6244 / 2297
error = .0000464
R = 2721 / 1001
error = -0.000000110
(George Sicherman)
R = 43623 / 16048
error = 8.24 × 10-7
(George Sicherman)

R = 49171 / 18089
error = 2.77 × 10-10
(Michael Moffitt)
R = 222630 / 81901
error = 4.59 × 10-10
(Dave Langers)
R = 714340 / 262791
error = 6.63 × 10-11
(Michael Moffitt)

ϕ Using Resistors 1-n

R = 1
error = -.618
R = 2 / 3
error = -.951
R = 3 / 2
error = .118
R = 44 / 27
error = .0116
R = 60 / 37
error = .00359

R = 34 / 21
error = .00101
R = 7998 / 4943
error = -.0000117
R = 29754 / 18389
error = -0.00000147
(George Sicherman)
R = 2584 / 1597
error = -1.75 × 10-7
(Jon Palin)

R = 526772 / 325563
error = 1.25 × 10-9
(George Sicherman)
R = 46368 / 28657
error = -5.45 × 10-10
(Dave Langers)
R = 5210550 / 3220297
error = 4.06 × 10-11
(Michael Moffitt)

√2 Using Resistors 1-n

R = 1
error = -.414
R = 2 / 3
error = -.748
R = 4 / 3
error = -.0809
R = 44 / 31
error = .00514
R = 24 / 17
error = -.00245
R = 140 / 99
error = -.0000721
R = 6190 / 4377
error = -.00000292

R = 6685 / 4727
error = .00000264
(George Sicherman)
R = 32814 / 23203
error = 1.17 × 10-7
(George Sicherman)
R = 14845 / 10497
error = 2.26 × 10-8
(Jon Palin)
R = 86523 / 61181
error = 6.61 × 10-10
(Dave Langers)
R = 1587920 / 1122829
error = 2.30 × 10-11
(Michael Moffitt)

George Sicherman suggested we approximate only using odd resistors:

π Using Odd Resistors 1-n

R = 1
error = -2.14
R = 4
error = .858
R = 23 / 8
error = -.267
(George Sicherman)
R = 161 / 51
error = .0153
(George Sicherman)
R = 518 / 165
error = -.00220
(George Sicherman)
R = 2077 / 661
error = .000616
(George Sicherman)
R = 2108 / 671
error = -.0000129
(George Sicherman)

R = 355 / 113
error = 2.67 × 10-7
(Dave Langers)
R = 179608 / 57171
error = 1.12 × 10-7
(Dave Langers)
R = 6000687 / 1910078
error = -6.59 × 10-9
(Dave Langers)
R = 12493715 / 3976873
error = -2.66 × 10-10
(Dave Langers)

e Using Odd Resistors 1-n

R = 1
error = -1.72
R = 4
error = 1.282
R = 23 / 8
error = .157
(George Sicherman)
R = 11 / 4
error = .0317
(George Sicherman)
R = 280 / 103
error = .000165
(George Sicherman)
R = 6349 / 2335
error = .000776
(George Sicherman)
R = 12360 / 4547
error = -.00000604
(George Sicherman)

R = 220635 / 81167
error = 2.70 × 10-6
(Dave Langers)
R = 483335 / 177809
error = 1.48 × 10-7
(Dave Langers)
R = 14240785 / 5238892
error = 1.43 × 10-8
(Dave Langers)
R = 26607831 / 9788474
error = -2.60 × 10-10
(Dave Langers)

ϕ Using Odd Resistors 1-n

R = 1
error = -.618
R = 3 / 4
error = -.868
(George Sicherman)
R = 2
error = .382
(George Sicherman)
R = 15 / 9
error = -.0625
(George Sicherman)
R = 423 / 260
error = .00889
(George Sicherman)
R = 13980 / 8641
error = -.000166
(George Sicherman)
R = 65130 / 40253
error = -.0000179
(George Sicherman)

R = 31205 / 19286
error = -.0000209
(Dave Langers)
R = 677286 / 418585
error = -2.97 × 10-6
(Dave Langers)
R = 2507655 / 1549816
error = 2.30 × 10-8
(Dave Langers)
R = 278284266 / 171989135
error = -7.31 × 10-10
(Dave Langers)

√2 Using Odd Resistors 1-n

R = 1
error = -.414
R = 3 / 4
error = -.664
(George Sicherman)
R = 8 / 9
error = -.525
(George Sicherman)
R = 120 / 79
error = .105
(George Sicherman)
R = 1260 / 887
error = .00631
(George Sicherman)
R = 7287 / 5150
error = .000738
(George Sicherman)

R = 3015 / 2132
error = -.0000485
(George Sicherman)
R = 24885 / 17597
error = -.0000521
(Dave Langers)
R = 150195 / 106204
error = -1.29 × 10-6
(Dave Langers)
R = 4411352 / 3119297
error = -3.93 × 10-8
(Dave Langers)
R = 27720 / 19601
error = -1.84 × 10-9
(Dave Langers)

Umut Uludag suggested we approximate using n identical integer resistors:

π Using n Identical Integer Resistors

R = 3
error = -.14
(Umut Uludag)
R = 3
error = -.14
(Umut Uludag)
R = 3
error = -.14
(Umut Uludag)
R = 16 / 5
error = .0584
(Umut Uludag)
R = 22 / 7
error = .00127
(Umut Uludag)
R = 22 / 7
error = .00127
(George Sicherman)
R = 22 / 7
error = .00127
(George Sicherman)
R = 22 / 7
error = .00127
(Dave Langers)

R = 22 / 7
error = .00127
(Dave Langers)
R = 22 / 7
error = .00127
(Dave Langers)
R = 355 / 113
error = 2.67 × 10-7
(Dave Langers)
R = 355 / 113
error = 2.67 × 10-7
(Dave Langers)

R = 355 / 113
error = 2.67 × 10-7
(Dave Langers)
R = 355 / 113
error = 2.67 × 10-7
(Dave Langers)
R = 355 / 113
error = 2.67 × 10-7
(Dave Langers/Michel Moffit)

e Using n Identical Integer Resistors

R = 3
error = .282
(Umut Uludag)
R = 5 / 2
error = -.218
(Umut Uludag)
R = 8 / 3
error = -.052
(Umut Uludag)
R = 11 / 4
error = .032
(Umut Uludag)
R = 8 / 3
error = -.052
(Umut Uludag)
R = 30 / 11
error = .00899
(Dave Langers)
R = 49 / 18
error = .00394
(Dave Langers)
R = 19 / 7
error = -.00400
(Dave Langers)

R = 125 / 46
error = -.000891
(Dave Langers)
R = 106 / 39
error = -.000333
(Dave Langers)
R = 280 / 103
error = .000165
(Dave Langers)
R = 492 / 181
error = -.0000498
(Dave Langers)

R = 193 / 71
error = .0000280
(Dave Langers)
R = 1264 / 465
error = -.00000265
(Dave Langers)
R = 1264 / 465
error = -.00000265
(Dave Langers)

ϕ Using n Identical Integer Resistors

R = 2
error = .382
(Umut Uludag)
R = 3 / 2
error = -.118
(Umut Uludag)
R = 5 / 3
error = .049
(Umut Uludag)
R = 8 / 5
error = -.018
(Umut Uludag)
R = 8 / 5
error = -.018
(Umut Uludag)
R = 21 / 13
error = -.00265
(Dave Langers)
R = 21 / 13
error = -.00265
(Dave Langers)
R = 34 / 21
error = .00101
(Dave Langers)

R = 55 / 34
error = -.000387
(Dave Langers)
R = 89 / 55
error = .000148
(Dave Langers)
R = 144 / 89
error = -.0000565
(Dave Langers)
R = 233 / 144
error = .0000216
(Dave Langers)

R = 377 / 233
error = -8.24 × 10-6
(Dave Langers)
R = 610 / 377
error = 3.15 × 10-6
(Dave Langers)
R = 987 / 610
error = -1.20 × 10-6
(Dave Langers)

√2 Using n Identical Integer Resistors

R = 1
error = -.414
(Umut Uludag)
R = 3 / 2
error = .086
(Umut Uludag)
R = 4 / 3
error = -.0809
(Umut Uludag)
R = 4 / 3
error = -.0809
(Umut Uludag)
R = 7 / 5
error = -.0142
(Umut Uludag)
R = 7 / 5
error = -.0140
(Dave Langers)
R = 24 / 17
error = -.00245
(Dave Langers)

R = 7 / 5
error = -.00245
(Dave Langers)
R = 41 / 29
error = -.000420
(Dave Langers)
R = 99 / 70
error = .0000722
(Dave Langers)
R = 140 / 99
error = -.0000721
(Dave Langers)
R = 140 / 99
error = -.0000721
(Dave Langers)

R = 239 / 169
error = -.0000124
(Dave Langers)
R = 239 / 169
error = -.0000124
(Dave Langers)
R = 816 / 577
error = -.00000212
(Dave Langers)

Here are the best known approximations using any n integer resistors:

π Using Any n Integer Resistors

R = 3
error = -.142
R = 60 / 19
error = .0163
R = 12180 / 3877
error = .0000117
(Jon Palin)
R = 10292453220 / 3276189613
error = 9.77 × 10-12
(Jon Palin)

e Using Any n Integer Resistors

R = 3
error = .282
R = 87 / 32
error = .000468
R = 457794 / 168413
error = 1.44 × 10-8
(Jon Palin)
R = 8104999681386 / 2981662753483
error = -2.31 × 10-16
(Jon Palin) (Alex Rower)

ϕ Using Any n Integer Resistors

R = 2
error = .382
R = 8 / 5
error = -.0180
R = 144 / 89
error = -.0000565
R = 8261955 / 5106169
error = -1.12 × 10-9
(Jon Palin)

√2 Using Any n Integer Resistors

R = 1
error = -.414
R = 10 / 7
error = .0144
R = 1410 / 997
error = .0000292
(Jon Palin)
R = 19338150 / 13674137
error = 6.23 × 10-11
(Jon Palin)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/1/13.